Arnesh Sujanani – Optimization Online

Optimal Nonergodic Primal-Dual Complexity of Efficient Inexact Parameter-Free Augmented Lagrangian Methods

Published: 2026/08/04, Updated: 2026/08/10

Augmented Lagrangian (AL) methods are a classical framework for constrained optimization, but for directly verifiable approximate KKT points, known first-order complexity bounds for standard inexact AL methods are suboptimal, while the best known proximal augmented Lagrangian (PAL) bounds retain an additional logarithmic factor. We consider linearly constrained convex composite problems with a smooth convex term … Read more

Optimal diagonal preconditioning beyond worst-case conditioning: theory and practice of omega scaling

Published: 2025/09/27, Updated: 2026/05/14

Convex and Nonsmooth Optimization, Nonsmooth Optimization, Optimization in Data Science nonconvex nonsmooth optimization, optimal diagonal preconditioner, preconditioning, pseudoconvex function, subgradient method

Preconditioning is essential in many areas of mathematics, and in particular is a fundamental tool for accelerating iterative methods for solving linear systems. In this work, we study optimal diagonal preconditioning under two distinct notions of conditioning: the classical worst-case \(\kappa\)-condition number and the averaging-based \(\omega\)-condition number. We observe that \(\omega\)-optimal preconditioning generally outperforms \(\kappa\)-optimal … Read more

A user manual for cuHALLaR: A GPU accelerated low-rank semidefinite programming Solver

Published: 2025/08/21

Nonlinear Optimization, Optimization Software and Modeling Systems, Semi-definite Programming augmented Lagrangian, Burer-Monteiro factorization, frank-wolfe, gpu, large scale semidefinite programming, low-rank factorization, SDP solver

We present a Julia-based interface to the precompiled HALLaR and cuHALLaR binaries for large-scale semidefinite programs (SDPs). Both solvers are established as fast and numerically stable, and accept problem data in formats compatible with SDPA and a new enhanced data format taking advantage of Hybrid Sparse Low-Rank (HSLR) structure. The interface allows users to load … Read more

Asymptotically Fair and Truthful Allocation of Public Goods

Published: 2025/06/06

Arnesh Sujanani

Convex and Nonsmooth Optimization, Game Theory, Nonlinear Optimization admm, asymptotically fair, differential privacy, game theory

We study the fair and truthful allocation of m divisible public items among n agents, each with distinct preferences for the items. To aggregate agents’ preferences fairly, we focus on finding a core solution. For divisible items, a core solution always exists and can be calculated by maximizing the Nash welfare objective. However, such a … Read more

cuHALLaR: A GPU accelerated low-rank augmented Lagrangian method for large-scale semidefinite programming

Published: 2025/05/19, Updated: 2025/10/23

Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming, Optimization Software and Modeling Systems augmented Lagrangian, conic and semidefinite programming, convex and nonconvex optimization, frank-wolfe, gpu, low-rank factorization

This paper introduces cuHALLaR, a GPU-accelerated implementation of the HALLaR method proposed in Monteiro et al. 2024 for solving large-scale semidefinite programming (SDP) problems. We demonstrate how our Julia-based implementation efficiently uses GPU parallelism through optimization of simple, but key, operations, including linear maps, adjoints, and gradient evaluations. Extensive numerical experiments across three problem classes—maximum … Read more

Efficient parameter-free restarted accelerated gradient methods for convex and strongly convex optimization

Published: 2024/10/05, Updated: 2024/10/11

Convex and Nonsmooth Optimization, Convex Optimization, Nonlinear Optimization complexity, convex optimization, first-order methods, parameter free methods, restarted accelerated method, strongly convex optimization

This paper develops a new parameter-free restarted method, namely RPF-SFISTA, and a new parameter-free aggressive regularization method, namely A-REG, for solving strongly convex and convex composite optimization problems, respectively. RPF-SFISTA has the major advantage that it requires no knowledge of both the strong convexity parameter of the entire composite objective and the Lipschitz constant of … Read more

A low-rank augmented Lagrangian method for large-scale semidefinite programming based on a hybrid convex-nonconvex approach

Published: 2024/01/22, Updated: 2024/04/01

Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming, Nonlinear Optimization accelerated proximal point algorithms, adaptive algorithms, augmented Lagrangian, conic and semidefinite programming, Frank-Wolfe algorithm, global convergence rates, iteration complexity, low-rank optimization

This paper introduces HALLaR, a new first-order method for solving large-scale semidefinite programs (SDPs) with bounded domain. HALLaR is an inexact augmented Lagrangian (AL) method where the AL subproblems are solved by a novel hybrid low-rank (HLR) method. The recipe behind HLR is based on two key ingredients: 1) an adaptive inexact proximal point method … Read more

An adaptive superfast inexact proximal augmented Lagrangian method for smooth nonconvex composite optimization problems

Published: 2022/07/25, Updated: 2022/10/03

Constrained Nonlinear Optimization, Nonsmooth Optimization accelerated first-order methods, Adaptive methods, augmented lagrangian method, iteration complexity, linearly-constrained nonconvex composite optimization, smooth weakly convex function

This work presents an adaptive superfast proximal augmented Lagrangian (AS-PAL) method for solving linearly-constrained smooth nonconvex composite optimization problems. Each iteration of AS-PAL inexactly solves a possibly nonconvex proximal augmented Lagrangian (AL) subproblem obtained by an aggressive/adaptive choice of prox stepsize with the aim of substantially improving its computational performance followed by a full Lagrangian … Read more